Theorem 19.23vv 951

Description: Theorem 19.23 of [Margaris] p. 90 extended to two variables.

Assertion

Ref	Expression
19.23vv	⊢ (∀x∀y(φ → ψ) ↔ (∃x∃yφ → ψ))

Distinct variable group(s):   ψ,x   ψ,y

Proof of Theorem 19.23vv

Step	Hyp	Ref	Expression
1		19.23v 950	. . 3 ⊢ (∀y(φ → ψ) ↔ (∃yφ → ψ))
2	1	bial 695	. 2 ⊢ (∀x∀y(φ → ψ) ↔ ∀x(∃yφ → ψ))
3		19.23v 950	. 2 ⊢ (∀x(∃yφ → ψ) ↔ (∃x∃yφ → ψ))
4	2, 3	bitr 151	1 ⊢ (∀x∀y(φ → ψ) ↔ (∃x∃yφ → ψ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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